Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 259^\circ.$
Since the cosine function has period $360^\circ,$
\[\cos 259^\circ = \cos (259^\circ - 360^\circ) = \cos (-101^\circ).\]And since the cosine function is even, $\cos (-101^\circ) = \cos 101^\circ,$ so $n = \boxed{101}.$